Automorphism groups of Gaussian Bayesian networks

Abstract

In this paper, we extend earlier work on groups acting on Gaussian graphical models to Gaussian Bayesian networks and more general Gaussian models defined by chain graphs with no induced subgraphs of the form $i\to j-k$. We fully characterise the maximal group of linear transformations which stabilises a given model and we provide basic statistical applications of this result. This includes equivariant estimation, maximal invariants for hypothesis testing and robustness. In our proof, we derive simple necessary and sufficient conditions on vanishing subminors of the concentration matrix in the model. The computation of the group requires finding the essential graph. However, by applying Stúdeny’s theory of imsets, we show that computations for DAGs can be performed efficiently without building the essential graph.